Alternating Energy Minimization Methods for Multi-term Matrix Equations
Abstract
We develop computational methods for approximating the solution of a linear multi-term matrix equation in low rank. We follow an alternating minimization framework, where the solution is represented as a product of two matrices, and approximations to each matrix are sought by solving certain minimization problems repeatedly. The solution methods we present are based on a rank-adaptive variant of alternating energy minimization methods that builds an approximation iteratively by successively computing a rank-one solution component at each step. We also develop efficient procedures to improve the accuracy of the low-rank approximate solutions computed using these successive rank-one update techniques. We explore the use of the methods with linear multi-term matrix equations that arise from stochastic Galerkin finite element discretizations of parameterized linear elliptic PDEs, and demonstrate their effectiveness with numerical studies.
Cite
@article{arxiv.2006.08531,
title = {Alternating Energy Minimization Methods for Multi-term Matrix Equations},
author = {Kookjin Lee and Howard C. Elman and Catherine E. Powell and Dongeun Lee},
journal= {arXiv preprint arXiv:2006.08531},
year = {2020}
}